How "chol" and "qz" MATLAB algorithms are utilised in the "eig" MATLAB function?

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Zahraa
Zahraa le 13 Mar 2024
Commenté : Bruno Luong le 14 Mar 2024
The MATLAB function "eig" when used this way: [V,D] = eig(A,B), gives us the eigenvalues and eigenvectors of an eigenvalue problem A*V = B*V*D, where D is the diagonal matrix of eigenvalues and matrix V columns are the corresponding right eigenvectors.
In the documentation, it is menioned that the function "eig" uses the algorithms "chol" (Cholesky factorization), and "qz" (QZ factorization for generalized eigenvalues / generalized Schur decomposition).
However, when reading about the methods of Cholesky factorization and QZ factorization for generalized eigenvalues, the first method only decomposes a matrix into a product of where L is a lower triangular matrix, and the second method computes the eigenvalues and what about the eigenvectors?
So, for this reason, I don't understand how using only these algorithms could return us both eigenvectos and eigenvalues of an eigenvalue problem.

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Bruno Luong
Bruno Luong le 13 Mar 2024
Modifié(e) : Bruno Luong le 13 Mar 2024
See stage 3 described in https://netlib.org/lapack/lug/node56.html or somewhere in this file
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Zahraa
Zahraa le 14 Mar 2024
Modifié(e) : Zahraa le 14 Mar 2024
Thank you. Thats right. MATLAB uses LAPACK in some linear algebra functions such as eig . In an answer here, it is mentioned that MATLAB function eig checks for symmetry, and then uses an appropriate call to LAPACK.
In a documentation, it is mentioned that: The primary tool for the solution of generalized matrix eigenvalue problems, Ax = λBx, is the function qz(A,B). MATLAB 5.3 has only one version of the QZ algorithm. It always uses complex arithmetic and produces complex results, even if the input matrices are real. LAPACK provides a real QZ algorithm for real matrices, so we can now have qz(A,B,'real') and qz(A,B,'complex').
Back to eig documentation, when it is said that eig uses the algorithms of the functions "chol" or "qz", why it doesn't also mention that it uses other functions in LAPACK, because in the stage 3 of the link you sent, the eigenvectors are computed using xTGEVC and xTGEVC solvers of LAPACK. So, "qz" does give us the eigenvalues, but alone doesn't give eigenvectors. And "chol" only decomposes a matrix as I wrote in my question.
There must be information missing in the section of algorithms of eig function of its documentaion?
Then, also:
In a documentation, it is mentioned that: The extensive list of functions now available with LAPACK means that MATLAB's space saving general-purpose codes can be replaced by faster, more focused routines. There are now 16 different code paths underlying the eig function, depending on whether there are one or two arguments, whether the arguments are real or complex, whether the problem is symmetric and whether the eigenvectors are requested.
This made me think, what do they mean with eig function having "16 different code paths".
And just a little more, here it is mentioned that "chol" uses some of the LAPACK routines.
Bruno Luong
Bruno Luong le 14 Mar 2024
Ran in:
MATLAB documentation mentions only qz and chol because it has the user switch of eig. It doesn't mention other stage because it judges most users do not care about knowing such thing. ou can consider it as missing if you want.
I don't know exactly how MATLAB uses chol.
All I know is it can transform generalized eigen decomposition of sdp matrix (which has specific algorithm) with Cholesky decomposition:
% Generate random sdp matrices A and B
A=rand(5);
A=A'*A;
B=rand(5);
B=B'*B;
eig(A,B)
ans = 5×1
0.0622 0.1110 1.2352 3.8498 7.6685
BB=chol(B); C=BB'\(A/BB);
C = (C + C')/2; % spd C
eig(C)
ans = 5×1
0.0622 0.1110 1.2352 3.8498 7.6685

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